M/GI/1 queues with services of both positive and negative customers

2004 ◽  
Vol 41 (04) ◽  
pp. 1157-1170 ◽  
Author(s):  
Yijun Zhu ◽  
Zhe George Zhang
Keyword(s):  
Generating Functions ◽  
Time Distribution ◽  
Opposite Sign ◽  
Probability Distributions ◽  
Waiting Time Distribution ◽  
Negative Customers ◽  
Special Cases ◽  
Stationary Waiting Time ◽  
Original Concept ◽  
Negative Arrivals

We consider an M/GI/1 queue with two types of customers, positive and negative, which cancel each other out. The server provides service to either a positive customer or a negative customer. In such a system, the queue length can be either positive or negative and an arrival either joins the queue, if it is of the same sign, or instantaneously removes a customer of the opposite sign at the end of the queue or in service. This study is a generalization of Gelenbe's original concept of a queue with negative customers, where only positive customers need services and negative customers arriving at an empty system are lost or need no service. In this paper, we derive the transient and the stationary probability distributions for the major performance measures in terms of generating functions and Laplace transforms. It has been shown that the previous results for the system with negative arrivals of zero service time are special cases of our model. In addition, we obtain the stationary waiting time distribution of this system in terms of a Laplace transform.

M/GI/1 queues with services of both positive and negative customers

2004 ◽  
Vol 41 (4) ◽  
pp. 1157-1170 ◽  
Author(s):  
Yijun Zhu ◽  
Zhe George Zhang
Keyword(s):  
Generating Functions ◽  
Time Distribution ◽  
Opposite Sign ◽  
Probability Distributions ◽  
Waiting Time Distribution ◽  
Negative Customers ◽  
Special Cases ◽  
Stationary Waiting Time ◽  
Original Concept ◽  
Negative Arrivals

We consider an M/GI/1 queue with two types of customers, positive and negative, which cancel each other out. The server provides service to either a positive customer or a negative customer. In such a system, the queue length can be either positive or negative and an arrival either joins the queue, if it is of the same sign, or instantaneously removes a customer of the opposite sign at the end of the queue or in service. This study is a generalization of Gelenbe's original concept of a queue with negative customers, where only positive customers need services and negative customers arriving at an empty system are lost or need no service. In this paper, we derive the transient and the stationary probability distributions for the major performance measures in terms of generating functions and Laplace transforms. It has been shown that the previous results for the system with negative arrivals of zero service time are special cases of our model. In addition, we obtain the stationary waiting time distribution of this system in terms of a Laplace transform.

scholarly journals Single server queues with modified service mechanisms

1962 ◽  
Vol 2 (4) ◽  
pp. 499-507 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Time Dependent Problem ◽  
Stationary Waiting Time ◽  
Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (2) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

THE MX/D/c BATCH ARRIVAL QUEUE

2005 ◽  
Vol 19 (3) ◽  
pp. 345-349 ◽  
Author(s):  
Geert Jan Franx
Keyword(s):  
Explicit Expression ◽  
Waiting Time ◽  
Probabilistic Analysis ◽  
Generating Functions ◽  
Time Distribution ◽  
Laplace Transforms ◽  
Batch Arrival ◽  
Waiting Time Distribution

A surprisingly simple and explicit expression for the waiting time distribution of the MX/D/c batch arrival queue is derived by a full probabilistic analysis, requiring neither generating functions nor Laplace transforms. Unlike the solutions known so far, this expression presents no numerical complications, not even for high traffic intensities.

Waiting-Time Percentiles in the Multi-server Mx/G/c Queue with Batch Arrivals

1987 ◽  
Vol 1 (1) ◽  
pp. 75-96 ◽  
Author(s):  
A. M. Eikeboom ◽  
H. C. Tijms
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Linear Interpolation ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Batch Arrivals ◽  
Special Cases ◽  
Moment Approximation ◽  
Multi Server ◽  
Validation Experiments

This paper deals with the MX/G/c queue. Using analytical results for the special cases of the MX/M/c queue and the MX/D/c queue, a two-moment approximation is proposed for the waiting-time percentiles in the general case. This approximation is based on a linear interpolation with respect to the squared coefficient of variation of the service time distribution. Validation experiments indicate that this approximation performs quite well for practical purposes. In particular, the practically important percentiles in the tail of the waiting-time distribution are approximated extremely well.

scholarly journals The Workload in the M/G/1 Queue with Work Removal

1996 ◽  
Vol 10 (2) ◽  
pp. 261-277 ◽  
Author(s):  
Richard J. Boucherie ◽  
Onno J. Boxma
Keyword(s):  
Poisson Process ◽  
Waiting Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Negative Customers ◽  
Workload Distribution

We consider an M/G/1 queue with the special feature of additional negative customers, who arrive according to a Poisson process. Negative customers require no service, but at their arrival a stochastic amount of work is instantaneously removed from the system. We show that the workload distribution in this M/G/1 queue with negative customers equals the waiting time distribution in a GI/G/1 queue with ordinary customers only; the effect of the negative customers is incorporated in the new arrival process.

COMPUTATIONAL ANALYSIS OF STATIONARY WAITING-TIME DISTRIBUTIONS OF GIX/R/1 AND GIX/D/1 QUEUES

2005 ◽  
Vol 19 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Mohan L. Chaudhry ◽  
Dae W. Choi ◽  
Kyung C. Chae
Keyword(s):  
Closed Form ◽  
Waiting Time ◽  
Time Distribution ◽  
Computational Analysis ◽  
Rational Functions ◽  
Waiting Time Distribution ◽  
Generalized Distributions ◽  
Analytic Expression ◽  
Stationary Waiting Time ◽  
Stieltjes Transforms

In this article, we obtain, in a unified way, a closed-form analytic expression, in terms of roots of the so-called characteristic equation of the stationary waiting-time distribution for the GIX/R/1 queue, where R denotes the class of distributions whose Laplace–Stieltjes transforms are rational functions (ratios of a polynomial of degree at most n to a polynomial of degree n). The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn) but also covers nonphase-type distributions such as deterministic (D). In the latter case, we get approximate results. Numerical results are presented only for (1) the first two moments of waiting time and (2) the probability that waiting time is zero. It is expected that the results obtained from the present study should prove to be useful not only for practitioners but also for queuing theorists who would like to test the accuracies of inequalities, bounds, or approximations.

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